Weighted composition operators on the Dirichlet space: boundedness and spectral properties

Boundedness of weighted composition operators W u,φ acting on the classical Dirichlet space D as W u,φ f=u(f∘φ) is studied in terms of the multiplier space associated to the symbol φ , i.e., M(φ)={u∈D:W u,φ is bounded on D} . A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of W u,φ in D whenever φ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen et al. (J. Funct. Anal. 265:1749–1777, 2013) to the context of the Dirichlet space.